One of the aims of this paper is to extend the following result of Auslander and Lichtenbaum [1]}, [2]}: if \(R\)  is a regular local ring, \(M, N\in \operatorname{mod}(R)\)  and \(M\otimes _RN\)  is torsion-free, then \(M\)  and \(N\)  are Tor-independent torsion-free modules. Our technique relies upon using local cohomology theory with respect to specialization-closed subsets [3]}.
